Units 1 : Matrices and Determinants 36 Pds.
Matrix as a rectangular arrangement of numbers, types of matrices, equality of matrices; addition; scalar multiplication and multiplication of matrices; linear combinations of matrices, non-comutativity and associativity of matrix multiplication; singular and non-singular matrices; linear equations in matrix notations; determinants; minors and cofactors of a determinant; expansion of a determinant, properties and elementary transformation of determinants; application of derminants in solution of equations and area of a triangle; Cramer's rule, adjoint and inverse of a matrix and its properties; finding the inverse by elementary row transformation. Application of matrices in solving simultaneous equations in three variables.
Units 2 : Vectors and its Application to Geometry 19 Pds.
Vectors as directed line segment, magnitude and direction of a vector, equal vectors, unit vector, zero vector, position vector of a point, components of a vector, vectors in two and three dimentions; addition of a vectors; multiplication of a vector by a scalar, position vector of the point dividing a given segment in a given ratio, scalar (dot) product of vectors; prejection vector, cross product of two vectors; scalar triple product; vector triple product; application of vectors in the use of establishment of various geometrical results; work done = force x displacement. Proof of cosine rule, angle in a semi circle is a right angle; application of vector product in finding area of a triangle as 1/2 |a x b| and of a parallelogram as |a x b| proof of sine rule; application of scalar product in finding volume of a parallelopiped, coplanarity of vectors using scalar triple product; vector triple product.
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Unit 3: Three Dimensional Geometry
Decomposition of a vector into three non-coplanar directions as a base in 3-D, direction ratios and direction cosines of any vector; angle between two vectors; distance between two points; condition for the intersection of two lines; shortest distance between two lines, equation of a plane containing a given point and normal to a given direction, angle between two planes; angle between a line and a plane; distance of a point from a plane equation of any plane passing through the intersection of the two planes; equation of a sphere in the from |r-c| =a; and equation of sphere in the from |r-a| 2 + |r-b| 2 =c 2 where a and b are the position vectors of the extremities or the diameter and c is a constant.
Note : The carterstian equivalent of the vector treatment is to be stated side by side.
Unit 4: Differential Calculus 48 Pds.
Concept of real function, its domain and range, graph of a function, composition of functions.
Meaning of x - a, x - a-
Meaning of lim f(x), lim f(x), lim f(x) x - a, x - a+ x - a-
Fundamental theorems on limits.
Proof of the following (without treatment)
lim xn-an -------- = na n-1 (a > 0) x-a
lim sin x = 0, lim x = 1, lim sin x = 1 (0
lim log (1+ x) = 1, lim ex - 1
-------------- -------- = 1,
x - 0 x x - 0 x
Continuity of a function at a point, over an open/closed
interval; sum, product and quotient of continuous
functions, continuity of polynomial, exponential,
logarithmic and inverse trigonometric functions,
continuity of a composite function.
Derivative of a function, its geometrical and physical
significance, relationship between continuity and
differentiability.
Derivatives of x n, sin x, cos x, tan x from first
principles, theorems relation to the derivatives of the
sum, difference, product and quotient of functions,
derivative of trigonometric functions, inverse
trigonometric functions, logarithmic function and
exponential function, differentiation of implicit
functions, logarithmic differentiation, derivative of
functions expressed in parametric from, derivatives of
higher order.
Application of derivatives: motion in a straight line,
motion under gravity, rate of change of quantities,
increasing and decreasing functions and sign of the
derivative, maxima and minima (absolute, local), Rolle's
theorem and mean value
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theorem (without proof), curve sketching, meaning of
differential, errors and approximations.
Unit 5: Integral Calculus
53 Pds.
Integration as the inverse of differentiation,
indefinite integral or antiderivative; properties of
integrals.
f (f (x) + g (x) ) dx = f g (x) dx
f f(x) dx = c f f (x) dx
Fundamental integrals involving algebraic, trigonometric
and exponential functions; integration by substitution,
integral of the type
f d x f d x f d x f dx
------- ------- v x2 + a2 v x2-a2
x2 + a2 a2-x2
f dx f (p x+ q)
--------- -------------= dx
ax2 + bx + c ax2 + bx + c
Integration by parts; Integrals of the type
f sib -1 x dx, f e ax sin bx dx, f v x2 + a2 dx,
f v ax2 + bx + c dx,
f (px + q) v ax2 + bx +c dx
Integrals of the type
f dx f dx f sinn x cos m x dx
---------- ------------
a+ b cos x , a+ b sin x
Partial fractions and their use in integration.
Definite Integral
Definition of definite integral as the limit of a sum illustrated with the help of simple examples, fundamental theorem of calculus, evaluation of definite integrals, transformations of definite integrals by substitution, properties of definite integrals.
bf f (x) dx = - af f (x) dx, a b
afb f (x) dx = cfa f (x ) dx + bfc s (x) dx,
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of2a f (x) dx = ofa f (x ) dx + of f ( 2a- x) dx,
fb f (x) dx = fo f (a+b-x) dx,
afo f(x) dx = afo f (a-x) dx,
af2a f(x) dx = 2 fo f (x) dx, if f (2a-x)= f(x) and
=0, if f(2a-x) = -f(x)
af_a f (x) dx =2 ofa f(x) dx, if f(x) is even function of x and
= 0, if f (x) is odd function of x.
Evaluation of some integrals using the above properties, definite integral and area bonded by a curve, circle, parabola, ellipse and hyperbola in standard form between two ordinates and x- axis, area between two curves.
Unit 6 : Differential Equations 19 Pds.
Differential equation, order and degree, general and particular solution, formation of a differential equation whose general solution is given. Solution is given. Solution of a differential equation by the method of `Variable Separable', homogeneous equations and their solution; solution of linear equations of first order of the type.
dy/dx + P (x)y = Q (x)
Unit 7 : Correlation and Regression 19 Pds.
Bivariate frequency distributions as arising from observations of two variables on the same unit of observation, marginal and conditional frequency distributions derived from a bivariable frequency distribution, concept of relationship between variables introduced as the dependence of conditional on the values of the conditioning variable; distinction between relationship and functional relationship.
Correlation analysis as the measurement of the strength of linear relationship between two quantitative variables and regression analysis as the method of prediction the values of one quantitative variable from those of the other quantitative variable.
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Definition and calculation of the correlation coefficient, positive and negative correlation, perfect correlation, use of the scatter diagram in interpreting the values of the correlation coefficient.
Calculation of the regression coefficient and the two lines of regression by the method of least squares, use of the lines of regression for prediction.
Unit 8 : Probability
Random experiment and the associated sample space (i.e. set of all outcomes); Events as subsets of the sample space, occurrence of an event; impossible event, mutually exclusive events.
Elementary events, equally likely elementary events; definition of probability of an event as the ratio of the number of favourable equally likely events to the total number of equally likely events, addition rule for mutually exclusive events. Combination of events through the operations "and", "not", and their set representation; probability of the events "A" or "B", "not A".
Conditional probability; independent events; independent experiments; calculation of probabilities of events associated with independent experiments ; applications of Bayes' theorem.
Random variable as a function on a sample space, (only random variables taking finite number of values to be considered).
Distribution of random variable derived from the probabilities of events on the sample space on which the random variable is defined.
Binomial distribution; examples of different random experiments giving rise to random variables with the binomial distribution.
Notes: (i) Calculators are not permitted in the examination hall in the subject of Mathematics and for that matter in any other subject;
(ii) The use of logarithm tables, however, is permitted in the examination hall;
(iii) In case any question in the paper of Mathematics requires long calculations, the students in such a situation are advised to work out calculations only upto the two decimal place irrespective of the instructions contained in the question paper.
Books Recommended
1. Mathematics Part -I NCERT Publication
2. Mathematics Part -II -do-
3. Mathematics Part -III -do-
4. A New Book of Mathematics,, Part-II M/s Arya Book Depot,
30, Naiwala, Karol Bagh,
New Delhi-110005.
5. Sr. School-Mathematics for Class XII- M/s Frank Bros. & Co.,
English 4675-A, Ansari Road,
Darya Ganj,
New Delhi-110002.
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6. Senior School Mathematics, Part-II M/s Goyal Bros. Prakashan (for Class XII-English) 11/1903, Chuna Mandi, Pahar Ganj, New Delhi-110055.
7. Modern Approach to Mathematics-English M/s Holy Faith Internati- onal Pvt. Ltd., 6,Bahadurshah Zafar Marg, New Delhi- 110002.
8. A Course in Mathematics for M/s Inter University Press Pvt. Ltd., (Class XII) (Vol. I & II)-English 30/7, Shakti Nagar, Delhi-110007
9. Comprehensive Mathematics for M/s Laxmi Publications, Class XII-English 7/12, Ansari Road, Darya- Ganj, New Delhi- 110002.
10. Modern Approach to Mathematics for M/s Modern Publishers, Class XII-English Gulab Bhavan, 6,Bahadurshah Zafar Marg, New Delhi-110002
11. New Course Mathematics XII-English M/s Pradeep Publications, Opp. Sitla Mandir Jallandhur-144008.
12. A Text Book of Mathematics (Vol. II) M/s Pitamber Publishing for class XII-English Co, 888, East Park Road, Karol Bagh, New Delhi-110005.
13. Fundamental Mathematics for M/s Sultan Chand & Sons, Class XII-English 23, Darya Ganj, New Delhi-110002.
14. Mathematics for Class XII-English M/s Tata MacGraw Hills Publishing Co., 4/12, Asaf Ali Road New Delhi-110002.
Note: The Hindi versions of the NCERT books are also available.
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7. PHYSICS (Code No. 042)
Objectives
Through this course in Physics, the learner should:
(i) develop competence to pursue science based professional
courses like engineering, and medicine, in future career;
(ii) get knowledge, understanding and application abilities
about different aspects of Physics;
(iii) strengthen foundations for further study of Physics;
(iv) develop an interest in the study of Physics as a
discipline;
(v) acquire necessary manipulative and experimental skills.
CLASS XI (Theory)